LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
The growth rate your wealth actually experiences is closer to the ordinary average return minus half the variance. That gap is volatility drag. A book can show a positive average and still compound slowly, or fail to compound, because returns multiply rather than add.
Why it matters
Every portfolio, active or passive, compounds. Ending wealth is the starting value times the product of one plus each period's return. People often treat a positive average as enough. It is not. Two paths with the same average can finish in different places if one path swings more. The quieter path keeps more of the average. The violent path gives part of that average back to the swings.
A Sharpe ratio, excess return per unit of variability, answers a different question than compounded growth. The setting that wins one contest can lose the other, and the gap shows up in the equity curve over long horizons.
The idea in plain words
Start from the account value after many periods. It equals the product of one plus each return. Products are awkward, so replace the whole product with a single growth rate, call it the geometric rate, raised to the number of periods. That single rate, compounded, lands on the same ending value as the messy product.
Take the natural log of both sides. A log turns a product into a sum and pulls the exponent down. Divide by the number of periods and the geometric rate sits inside a log that equals the average of the logs of one plus each return.
Higher powers of a small number shrink fast. Two percent squared is already tiny. Cubing it is smaller still. Dropping those higher powers is an approximation for small returns, not a casual rounding trick.
Apply that approximation on both sides. The average of the returns appears, which is the arithmetic mean. The average of the squared returns still has to be rewritten. Expand the usual variance formula, the average of squared gaps from the mean. After the algebra, the average of the squared returns equals the variance plus the square of the mean. Substitute that in.
For realistic returns those two squared means are close to each other, so their difference is near zero. A quick check the lecture suggests: square something like 0.03 and something like 0.02 and compare the gap. It is small. What remains is the result people use: geometric growth is about the arithmetic mean minus half the variance. In symbols, that is mu minus sigma squared over two. The second term is the drag.
How to use it
Use the formula as a lens, not as a forecast. If you have an average return and a variance, subtract half the variance and you have a rough compounded growth rate. If two sleeves have similar averages, prefer the one with less variance when the goal is long-run compounded wealth. If you add leverage, the same logic gets harsher. Leverage scales both the average and the swings, and the variance term is squared in the drag, so borrowed exposure punishes a volatile book even when the average stays positive.
When you tune a strategy, name the objective. Compounded growth, or a cap on how deep the account can fall, is a different job from a risk-adjusted score.
What usually goes wrong
The common mistake is to stop at the arithmetic average and call the strategy healthy. The derivation says the path matters. Another mistake is to treat the approximation as exact for huge single-period moves. The Taylor step assumed returns near zero. A crash-sized return makes the dropped terms less harmless, so the simple half-variance rule is a guide, not a substitute for multiplying the actual returns.
People also confuse the two optimizations. Improving a risk-adjusted ratio does not automatically maximize geometric growth, and maximizing growth without a survival constraint can still blow the account up if leverage is large. The lecture's point is that everyone who compounds is exposed to this gap, whether the book is tactical or passive.
Questions people ask
Is volatility drag a fee I pay to a broker?
No. It is the gap between the average return and the compounded rate, caused by multiplying uneven returns. No invoice shows it. It shows up when you compare the arithmetic mean with the growth rate that reproduces the ending wealth.
Does a positive average mean the account will grow?
Not by itself. The approximation says growth is the average minus half the variance. If the variance term eats the average, compounded growth can be flat or poor even though the average of the returns is positive.
Why does leverage make the drag worse?
Leverage magnifies each return. The variance, which enters the drag, rises faster than the average. The lecture's claim is that this nonlinear effect can wreck geometric growth even when expectancy stays positive. Survival to the long run is part of the problem, not a footnote.